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The mobility is proportional to the carrier relaxation time and inversly proportionnal to the carrier effective mass.
The electron mobilty is often greater than hole mobility because quite often, the electron effective mass is smaller than hole effective mass.
The relaxation times are often of the same order of magnitude for electrons and holes and therefore, they do not make too much difference.
In order to increase the speed of a device one has to choose materials with small electron and hole effective masses and long relaxation times i.e. where the electrons and holes do not have to experience too much collissions on crystal imperfections, impurities, ...i.e. one has to choose materials with high crystal quality.

@ Satyabrata Mahapatra
If you consider kinetic phenomena, effective mass is a measure of inertion of a charge carrier and it is directy related with the mobility.
The effective masses of electrons and holes are determined by the band structure of a mateial (very roughly speaking - by the curvature of the bands). Thus, if for example the curvature of the conduction band is smoother than that of the valence band, the electron's effective mass will be lower than that of the hole. BUT, this will not necessarily lead to higher electron mobility! In most real matrials the situation is complicated by the presence of different valleys, the charge carriers in which can scatter on each other. Moreover, the scattering of charge carriers on defects in a crystal can significantly reduce the mean free path and the elastic scattering time which also determines mobility - mobility decreases. In general, one has to consider both band structure and the presence of lattice defects.

In order to increase the mobility one has to produce very - very pure structures. I would advise you to read the papers on Si - inversion layers and MOS-structures studied by V.M. Pudalov and co-workers [Phys. Rev. B 50, 8039–8042 (1994) , Phys. Rev. B 51, 7038–7045 (1995)].

A more basic consideration of effective mass and mobility is given in Ziman "Principles of the theory of solids".

I'd like to add to what Taras said by clarifying that a higher curvature band suggests a lower effective mass, rather than a smoother one. Also, it's intuitive but not explicitly stated that for very short channel devices, the defects won't have as large of an effect on the mobility because electron motion becomes ballistic. Also, mobility can be dependent on the electric field applied. This is due to the valley scattering that Taras mentioned. At higher electric fields, carriers will have the energy to scatter into higher vallies. If these vallies have lower curvature, or higher effective mass, this can change the overall mobility.

The previous responses are correct. I would like to just say that it's not always the case that electron mobility is higher than hole mobility. One case is graphene where, because of the symmetric band structure, the electron and hole mobilities are the same. Also, it is well documented in the literature and applied universally in industry that adding strain to the channel material correctly can increase mobility.

one of the factors influnencing mobility is mass. it is not the same as rest mass of electrons as electrons in a lattice is influenced by a periodic potenial created by lattice constituents. therefore the curvature of a particular band defines the effective mass. if one is able to manipulate the bands it is possible to vary the effective mass and therefore mobility. e.g. high electron mobility transistrs (HEMT). additionally influence of external electric field has been well explained by Yearsley.

As well in n and p conductivity, it is always the electrons which move and conduct electricity.
For n conduction you have few carriers in the available sites and many empty states.
For p conduction you have many electrons at the given energy and few empty states.
The mobility comes from the scattering of a carrier from its initial state to a final state by a scattering potential. The scattering potential is the same for electrons and "holes". What changes is the energy of electrons
One can show that the mobility increases with the energy of electrons. Thus we can conclude that "n carriers" which are at higher energies than "p carriers" are more mobile.

The mobility is proportional to the carrier relaxation time and inversly proportionnal to the carrier effective mass.
The electron mobilty is often greater than hole mobility because quite often, the electron effective mass is smaller than hole effective mass.
The relaxation times are often of the same order of magnitude for electrons and holes and therefore, they do not make too much difference.
In order to increase the speed of a device one has to choose materials with small electron and hole effective masses and long relaxation times i.e. where the electrons and holes do not have to experience too much collissions on crystal imperfections, impurities, ...i.e. one has to choose materials with high crystal quality.

Many good responses above. I'd add one trick to get high mobilities in heterojunction transistors, which is to engineer band structures with the result that the channel only contains free carriers, but not ionised donors, reducing scattering.